Mathematics

Linear Algebra

449 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

Let $\Delta _0=\begin{bmatrix}a _{11} & a _{12}  & a _{13}\a _{21}  & a _{22} &a _{23} \ a _{31} & a _{32} & a _{33}\end{bmatrix}$ (where $\Delta _0 \neq  0$) and let $\Delta _1$ denote the determinant formed by the cofactors of elements of $\Delta _0$ and $\Delta _2$ denote the determinant formed by the cofactor at $\Delta _1$ and so on $\Delta _n$ denotes the determinant formed by the cofactors at $\Delta _{n-1}$ then the determinant value of $\Delta _{n}$ is

  1. $\Delta _0^{2n}$
  2. $\Delta _0^{2^n}$
  3. $\Delta _0^{n^2}$
  4. $\Delta _0^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Delta _1=\Delta ^2 _0,\Delta _2=\Delta ^2 _1=\Delta ^{2^2} _0$
$\therefore \Delta _n=\Delta ^{2n} _0$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

 For what value of
k, the matrix $A = \begin{bmatrix} 4 & 3 -k\\ 1 & 2 \end{bmatrix}$ is
not invertible?

  1. (a)$k = -5,$
  2. (b)$k=5,$
  3. (c)$k=0$
  4. (d)$k=3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A matrix is not invertible if its determinant is zero. Det(A) = (4 * 2) - (1 * (3 - k)) = 8 - 3 + k = 5 + k. Setting 5 + k = 0 gives k = -5.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If$A=\left[ \begin{matrix} 1 & -5 & 7 \ 0 & 7 & 9 \ 11 & 8 & 9 \end{matrix} \right] $ , then  trace of matrix $A$ is

  1. $17$
  2. $25$
  3. $3$
  4. $12$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $A=\left[ \begin{matrix} 1 & -5 & 7 \ 0 & 7 & 9 \ 11 & 8 & 9 \end{matrix} \right] $
$tr(A)=sum\ of\ diagonal\ matrix$
$tr(A)=1+7+9=17$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $\displaystyle :A= \left [ a _{ij} \right ]$ is a scalar matrix of order $\displaystyle :n\times n$ such that $\displaystyle :a _{ij}= k $ for all then trace of A is equal to

  1. $\displaystyle \:nk$
  2. $\displaystyle \:n+k$
  3. $\displaystyle \:n/k$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By definition of trace of a scalar matrix of order n, 
$tr(A)=a _{11}+ a _{22}+a _{33}+.....+a _{nn}$

$=k+k+....k $ (upto n times)

$tr(A)=nk$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

Let $A+2B=\begin{bmatrix} 1 & 2 & 0 \ 6 & -3 & 3 \ -5 & 3 & 1 \end{bmatrix}$ and $2A-B=\begin{bmatrix} 2 & -1 & 5 \ 2 & -1 & 6 \ 0 & 1 & 2 \end{bmatrix}$, then $tr(A)-tr(B)$ has the value equal to

  1. 0

  2. 1

  3. 2

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$A+2B=\begin{bmatrix} 1 & 2 & 0 \ 6 & -3 & 3 \ -5 & 3 & 1 \end{bmatrix}$         .....(i)

$2A-B=\begin{bmatrix} 2 & -1 & 5 \ 2 & -1 & 6 \ 0 & 1 & 2 \end{bmatrix}$         .....(ii)

$\Rightarrow 4A-2B=\begin{bmatrix} 4 & -2 & 10 \ 4 & -2 & 12 \ 0 & 2 & 4 \end{bmatrix}$      ....(iii)

Adding (i) and (iii), we get
$5A=\begin{bmatrix} 5 & 0 & 10 \ 10 & -5 & 15 \ -5 & 5 & 5 \end{bmatrix}$

$\Rightarrow A=\begin{bmatrix} 1 & 0 & 2 \ 2 & -1 & 3 \ -1 & 1 & 1 \end{bmatrix}$
So, $tr(A)=1$

Now, by eq(ii),
$B=\begin{bmatrix} 2 & 0 & 4 \ 4 & -2 & 6 \ -2 & 2 & 2 \end{bmatrix}-\begin{bmatrix} 2 & -1 & 5 \ 2 & -1 & 6 \ 0 & 1 & 2 \end{bmatrix}$

$\Rightarrow B=\begin{bmatrix} 0 & 1 & -1 \ 2 & -1 & 0 \ -2 & 1 & 0 \end{bmatrix}$
So, $tr(B)=-1$
Now, $tr(A)-tr(B)=1+1=2$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

Let $A$ be the $2\times2$ matrices given by $A=\left[a _{ij}\right]$ where $a _{ij} = \left{0,1,2,3,4\right}$ such that $a _{11} + a _{12} + a _{21} + a _{22} = 4$
Find the number of matrices $A$ such that the trace of $A$ is equal to 4

  1. 3

  2. 4

  3. 5

  4. 6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given $tr(A)=4$
$\Rightarrow a _{11}+a _{22}=4$

$a _{ ij }={ { 0,1,2,3,4}  }$

So, diagonal entries of A can be 0 and 4 , 4 and 0, 1 and 3, 3 and 1, 2 and 2,
Hence, 5 matrices are possible

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $A=\begin{bmatrix} 2 & 1 \ 4 & 1 \end{bmatrix}; B=\begin{bmatrix} 3 & 4 \ 2 & 3 \end{bmatrix}$ and $C=\begin{bmatrix} 3 & -4 \ -2 & 3 \end{bmatrix}$ then $tr(A)+tr\left( \dfrac { ABC }{ 2 }  \right) +tr\left( \dfrac { A{ \left( BC \right)  }^{ 2 } }{ 4 }  \right) +tr\left( \dfrac { A{ \left( BC \right)  }^{ 3 } }{ 8 }  \right) +......\infty $ =

  1. $6$
  2. $9$
  3. $12$
  4. $15$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The expression is a geometric series of traces. Given the matrices, the sum converges to a specific value based on the properties of the trace and matrix multiplication.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

Consider three matrices A= $ \begin{bmatrix} 2 & 1 \ 4 & 1 \end{bmatrix} $, $ B = \begin{bmatrix} 3 & 4 \ 2 & 3 \end{bmatrix} $ and $ C = \begin{bmatrix} 3 & -4 \ -2 & 3 \end{bmatrix} $ Then the value of the sum 
$ tr(A)+tr \cfrac {(ABC) } {2} +tr \cfrac {A( {BC})^2} {4}+ \cfrac {A( {BC})^3} {2}  +...+ \infty               $is

  1. 6

  2. 9

  3. 12

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer